Extending Maps in Hilbert Manifolds
Volume 60 / 2012
Abstract
Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if is a completely metrizable space of topological weight not greater than \alpha \geq \aleph_0, A is a closed set in X and f\colon X \to M is a map into a manifold M modelled on a Hilbert space of dimension \alpha such that f(X \setminus A) \cap \overline{f(\partial A)} = \emptyset, then for every open cover \mathcal U of M there is a map g\colon X \to M which is \mathcal U-close to f (on X), coincides with f on A and is an embedding of X \setminus A into M. If, in addition, X \setminus A is a connected manifold modelled on the same Hilbert space as M, and \overline{f(\partial A)} is a Z-set in M, then the above map g may be chosen so that g|_{X \setminus A} be an open embedding.